Related papers: On the Complexity of Toric Ideals
We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint…
We study the parameterized complexity of the problems of finding a maximum common (induced) subgraph of two given graphs. Since these problems generalize several NP-complete problems, they are intractable even when parameterized by strongly…
We provide a new complexity bound for the computation of grevlex Gr\"obner bases in the generic zero-dimensional case, relying on Moreno-Soc\'ias' conjecture. We first formalize a property of regular sequences that implies a well-known…
The computational complexity of polynomial ideals and Gr\"obner bases has been studied since the 1980s. In recent years, the related notions of polynomial subalgebras and SAGBI bases have gained more and more attention in computational…
There has been great interest in identifying tractable subclasses of NP complete problems and designing efficient algorithms for these tractable classes. Constraint satisfaction and Bayesian network inference are two examples of such…
We consider the problem of learning graphical models, also known as Markov random fields (MRFs) from temporally correlated samples. As in many traditional statistical settings, fundamental results in the area all assume independent samples…
Probabilistic inferences distill knowledge from graphs to aid human make important decisions. Due to the inherent uncertainty in the model and the complexity of the knowledge, it is desirable to help the end-users understand the inference…
In this paper we present a version of the general polynomial involutive algorithm for computing Janet bases specialized to toric ideals. The relevant data structures are Janet trees which provide a very fast search for a Janet divisor. We…
We propose a measure based upon the fundamental theoretical concept in algorithmic information theory that provides a natural approach to the problem of evaluating $n$-dimensional complexity by using an $n$-dimensional deterministic Turing…
We prove that finding a $k$-edge induced subgraph is fixed-parameter tractable, thereby answering an open problem of Leizhen Cai. Our algorithm is based on several combinatorial observations, Gauss' famous \emph{Eureka} theorem [Andrews,…
In this paper, we study a class of toric ideals obtained by using some geometric data of ADE trees which are the minimal resolution graphs of rational surface singularities. We compute explicit Gr\"obner bases for these toric ideals that…
Graph Neural Networks (GNNs) have made significant advances on several fundamental inference tasks. As a result, there is a surge of interest in using these models for making potentially important decisions in high-regret applications.…
In graph realization problems one is given a degree sequence and the task is to decide whether there is a graph whose vertex degrees match to the given sequence. This realization problem is known to be polynomial-time solvable when the…
Markov random fields area popular model for high-dimensional probability distributions. Over the years, many mathematical, statistical and algorithmic problems on them have been studied. Until recently, the only known algorithms for…
In the talk at the workshop my aim was to demonstrate the usefulness of graph techniques for tackling problems that have been studied predominantly as problems on the term level: increasing sharing in functional programs, and addressing…
A theory of numerical path-following in toric varieties was suggested in two previous papers. The motivation is solving systems of polynomials with real or complex coefficients. When those polynomials are not assumed 'dense', solving them…
We present a learning-based approach to computing solutions for certain NP-hard problems. Our approach combines deep learning techniques with useful algorithmic elements from classic heuristics. The central component is a graph…
We present a $p$-adic algorithm to recover the lexicographic Gr\"obner basis $\mathcal G$ of an ideal in $\mathbb Q[x,y]$ with a generating set in $\mathbb Z[x,y]$, with a complexity that is less than cubic in terms of the dimension of…
Regularization has become a primary tool for developing reliable estimators of the covariance matrix in high-dimensional settings. To curb the curse of dimensionality, numerous methods assume that the population covariance (or inverse…
In this paper, we study toric ideals associated with multichains of posets. It is shown that the comparability graph of a poset is chordal if and only if there exists a quadratic Gr\"obner basis of the toric ideal of the poset. Strong…